Distribution goodness-of-fit test device, consumable goods supply timing judgment device, image forming device, distribution goodness-of-fit test method and distribution goodness-of-fit test program

ABSTRACT

A distribution goodness-of-fit test device for testing whether measured data matches an estimated probability distribution has a counting section determination unit, a counting unit and a goodness-of-fit test unit. The counting section determination unit determines according to the number of the measured data, widths of counting sections for counting the measured data. The counting unit counts the numbers of data in the respective determined counting sections. Also, the goodness-of-fit test unit performs a goodness-of-fit test based on the numbers of data in the respective counting sections.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a distribution goodness-of-fit testdevice for judging based on approximation to a chi-square distributionwhether measured data follows a discrete distribution such as a binomialdistribution or a Poisson distribution

2. Description of the Related Art

Conventionally, as to whether or not data obtained by measurementfollows a discrete distribution such as a binomial distribution or aPoisson distribution, there is used a chi-square goodness-of-fit testfor testing the goodness of fit by approximation to a chi-squaredistribution. In this test method, as to whether or not the transactionof persons, cars, packets or the like arriving at constant intervalsfollows a multinomial distribution, the goodness of fit is tested byapproximation to the chi-square distribution.

A method of the goodness-of-fit test to the Poisson distribution byusing the chi-square distribution will be described by use of a specificexample. Table 1 shows the result of checking the arrival number ofcalls received by a certain telephone line on a weekday for 30 days.Table 2 is a table obtained by classifying this according to the arrivalnumber. TABLE 1 day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x 6 3 3 5 3 5 54 5 6 7 1 8 5 5 day 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 x 5 108 3 5 4 7 8 3 6 3 3 3 9 2

TABLE 2 arrival observed frequency × number frequency arrival numberexpected frequency 0 0 0 0.20213841 1 1 1 1.01069205 2 1 2 2.526730125 38 24 4.211216874 4 2 8 5.264021093 5 8 40 5.264021093 6 3 18 4.3866842447 2 14 3.133345889 8 3 24 1.95834118 9 1 9 1.087967322 10  1 100.543983661 11  0 0 0.247265301 12  0 0 0.103027209 total 30 150

In a conventional goodness-of-fit test of a Poisson distribution byusing a chi-square distribution, first, an arrival rate is estimated asΣ (arrival number×observed frequency)/(total observed frequency). Table2 shows values of multiplication of arrival numbers and observedfrequencies. Since the multiplication values added together make 150, itis divided by 30 of the total observed frequency, and the arrival rate λis estimated as 5. FIG. 1 shows a graph in which a Poisson distributionwith the arrival rate λ=5 and observed values actually observed areoverlapped with each other and are indicated. Incidentally, the arrivalrate is the average value of the arrival number of arrivals per unittime, and corresponds to the average value of the Poisson distribution.

Next, an expected frequency is obtained from the estimated arrival rate.When the arrival rate is estimated, the distribution shape of thePoisson distribution is determined from this arrival rate. The expectedfrequency is a value of an observed frequency on the estimated Poissondistribution. For example, in the observed data shown in Table 1,although the time when the arrival number is 4 is observed twice, theexpected frequency of the arrival number 4 on the Poisson distributionis 5.264021093. The expected frequency can be obtained by a followingmathematical expression 1. Table 2 shows the expected frequencies at therespective arrival numbers. $\begin{matrix}{{{total}\quad{observed}\quad{frequency} \times ( {\frac{1}{{\mathbb{e}}^{\lambda}} \cdot \frac{\lambda^{x}}{x!}} )} = {30( {\frac{1}{{\mathbb{e}}^{5}} \cdot \frac{5^{x}}{5!}} )}} & (1)\end{matrix}$

Next, a test statistic is obtained by using a mathematical expression(2) set forth below. Where, X_(i) denotes an observed frequency in acounting section i of the arrival number, and Ei denotes an expectedfrequency in the counting section i of the arrival number. The countingsections in which the expected frequency is 1 or more are not changed,and the counting sections in which the expected frequency is less than 1are combined into one, and the total number of the counting sections ismade m. $\begin{matrix}{\chi_{2}^{0} = {\sum\limits_{i = 1}^{m}\quad\frac{( {x_{i} - E_{i}} )^{2}}{E_{i}}}} & (2)\end{matrix}$

This test statistic is compared with a chi-square value χ²(m−2, α) witha degree of freedom of m−2, and when the test statistic is larger thanthe chi-square value χ²(m−2, α), it is judged that “the observed datadoes not follow the estimated Poisson distribution”, and when the teststatistic is smaller than the chi-square value χ²(m−2, α), it is judgedthat “it can not be said that the observed data does not follow theestimated Poisson distribution”. α represents a significant level.

In the foregoing example, the test statistic χ₀ ²=9.208023<χ²(8,0.05)=15.50731249, and the test result is such that “it can not be saidthat the observed data does not follow the Poisson distribution”.

Patent document 1 discloses a method of testing the goodness of fit of asoftware reliability growth curve by using the property of bugs ofcomputer programs as a sequence statistic and censoring data.

Patent document 2 proposes a method of judging whether or not adifference in observation condition has an influence on the way ofoccurrence of an event by comparing two data obtained by observationperformed the same number of times while the observation condition ischanged.

[Patent document 1] Japanese Patent No. 2693435

[Patent document 2] JP-A-2003-281116

Unless the expected frequency is 10 or more, the test statistic forperforming the approximation to the chi-square distribution becomesinsufficient in accuracy. Thus, it is necessary that the number of datais made the number of classifications×10 or more. However, in the casewhere the number of classifications is large, there arises a problemthat it takes time and cost to collect data. Further, since the observeddata is not uniformly obtained in the respective classificationsections, there is also a classification section in which even if thenumber of data is increased, the expected frequency does not becomelarge. In this case, it becomes necessary to perform a processing ofcombining the classification in which the expected frequency is small,and there arises a problem that it takes time to perform the processing.

Besides, as is apparent from the diagram of trains or buses, a normalarrival rate (an average value of the arrival number of arrivals perunit time) is not constant, but follows a inhomogeneous Poisson processin which the arrival rate varies. However, in the inhomogeneous Poissonprocess, the distribution shape itself is changed in accordance with thechange of the arrival rate, it has been impossible to apply thechi-square goodness-of-fit test as it is.

Patent documents 1 and 2 do not disclose techniques to solve theproblems as stated above.

SUMMARY OF THE INVENTION

The present invention has been made in view of the above circumstancesand provides a distribution goodness-of-fit test device in which a testpossible range of a goodness-of-fit test is widened, a consumable goodssupply timing judgment device, an image forming device, a distributiongoodness-of-fit test method, and a distribution goodness-of-fit testprogram.

According to an aspect of the present invention, the distributiongoodness-of-fit test device for testing whether measured data matches anestimated probability distribution includes a counting sectiondetermination unit which determines, according to the number of themeasured data, widths of counting sections for counting the measureddata, a counting unit which counts the numbers of data in the respectivedetermined counting sections, and a goodness-of-fit test unit whichperform a goodness-of-fit test using the numbers of data in therespective counting sections.

According to this invention, the widths of the counting sections forcounting the data are determined according to the number of the measureddata. Accordingly, even in the case where the number of the dataobtained by measurement is small, or even if there is a counting sectionin which the number of the obtained data is small, the widths of thecounting sections are changed so that the test with high accuracybecomes possible. Thus, the test possible range of the goodness-of-fittest can be widened.

According to another aspect of the invention, the distributiongoodness-of-fit test device for testing whether measured data matches anestimated probability distribution includes a counting sectiondetermination unit which determines, according to the estimatedprobability distribution, widths of counting sections for counting themeasured data, a counting unit which counts the numbers of data in therespective determined counting sections, and a goodness-of-fit test unitwhich perform a goodness-of-fit test using the numbers of data in therespective counting sections.

According to this invention, the widths of the counting sections inwhich the measured data are counted are determined according to themeasured probability distribution. When the widths of the countingsections are changed according to the estimated probability distributionso that for example, the probabilities of the respective countingsections on the estimated probability distribution become equalprobabilities, it becomes possible to perform the goodness-of-fit testirrespective of the distribution shape of the probability distribution.Thus, it also becomes possible to perform the goodness-of-fit test for aprobability distribution varying temporally, and the test possible rangeof the goodness-of-fit test can be widened.

According to another aspect of the invention, the distributiongoodness-of-fit test device for testing whether measured data matches anestimated probability distribution includes a counting sectiondetermination unit which determines, according to the estimatedprobability distribution and the number of the measured data,distribution widths of counting sections for counting the measured data,a counting unit which counts the numbers of data in the respectivedetermined counting sections, and a goodness-of-fit test unit whichperforms a goodness-of-fit test using the numbers of data in therespective counting sections.

According to this invention, the widths of the counting sections forcounting the data are determined according to the number of the measureddata and the estimated probability distribution. Since the widths of thecounting sections are changed according to the number of the measureddata, even in the case where the number of the data obtained bymeasurement is small, or even if there is a counting section in whichthe number of the obtained data is small, the widths of the countingsections are changed so that the test with high accuracy becomespossible. Besides, when the widths of the counting sections are changedaccording to the estimated probability distribution so that for example,the probabilities of the respective counting sections on the estimatedprobability distribution become equal probabilities, it becomes possibleto perform the goodness-of-fit test irrespective of the distributionshape of the probability distribution. Thus, it also becomes possible toperform the goodness-of-fit test for a probability distribution varyingtemporally, and the test possible range of the goodness-of-fit test canbe widened.

According to another aspect of the present invention, the consumablegoods supply timing judgment device includes the foregoing distributiongoodness-of-fit test device for performing a goodness-of-fit test from ameasured consumption rate of consumable goods per unit time and anaverage value of a past consumption rate of the consumable goods perunit time, and a control unit which calculates an estimated supply timefrom a ratio of the present consumption rate of the consumable goods tothe past consumption rate of the consumable goods and notifies anestimated time of supply in a case where it is judged by thegoodness-of-fit test that there is a significant difference between thepresent consumption rate and the past consumption rate.

It is judged by the goodness-of-fit test whether there is a significantdifference between the present consumption rate and the past consumptionrate, and in the case where it is judged that there is a significantdifference, the estimated supply time is calculated from the ratio ofthe present consumption rate of the consumable goods to the pastconsumption rate of the consumable goods, so that the estimated time ofsupply can be calculated with high accuracy.

According to another aspect of the present invention, the image formingdevice includes the foregoing consumable goods supply timing judgmentdevice.

According to this invention, since the consumable goods supply timingjudgment device is provided, the supply timing of toner, sheets or thelike is judged with high accuracy, and the supply timing can be notifiedto the user.

According to another aspect of the present invention, the distributiongoodness-of-fit test method for testing whether measured data matches anestimated probability distribution includes a counting sectiondetermination step of determining, according to the estimatedprobability distribution and the number of the measured data, widths ofcounting sections for counting the measured data, a counting step ofcounting the numbers of data in the respective determined countingsections, and a goodness-of-fit test step of performing agoodness-of-fit test using the numbers of data in the respectivecounting sections.

According to this invention, the widths of the counting sections forcounting the data are determined according to the number of the measureddata and the estimated probability distribution. Since the widths of thecounting sections are changed according to the number of the measureddata, even in the case where the number of the data obtained bymeasurement is small, or even if there is a counting section in whichthe number of the obtained data is small, the widths of the countingsections are changed so that the test with high accuracy becomespossible. Besides, when the widths of the counting sections are changedaccording to the estimated probability distribution so that for example,the probabilities of the respective counting sections on the estimatedprobability distribution become equal probabilities, it becomes possibleto perform the goodness-of-fit test irrespective of the distributionshape of the probability distribution. Thus, it also becomes possible toperform the goodness-of-fit test for a probability distribution varyingtemporally, and the test possible range of the goodness-of-fit test canbe widened.

According to another aspect of the present invention, a storage mediumreadable by a computer stores a distribution goodness-of-fit testprogram of instructions executable by the computer to perform a functionfor testing whether measured data matches an estimated probabilitydistribution, the function comprising the steps of determining,according to the estimated probability distribution and the number ofthe measured data, widths of counting sections for counting the measureddata, counting the numbers of data in the respective determined countingsections, and performing a goodness-of-fit test using the numbers ofdata in the respective counting sections.

According to this invention, the widths of the counting sections forcounting the data are determined according to the number of the measureddata and the estimated probability distribution. Since the widths of thecounting sections are changed according to the number of the measureddata, even in the case where the number of the data obtained bymeasurement is small, or even if there is a counting section in whichthe number of the obtained data is small, the widths of the countingsections are changed so that the test with high accuracy becomespossible. Besides, when the widths of the counting sections are changedaccording to the estimated probability distribution so that for example,the probabilities of the respective counting sections on the estimatedprobability distribution become equal probabilities, it becomes possibleto perform the goodness-of-fit test irrespective of the distributionshape of the probability distribution. Thus, it also becomes possible toperform the goodness-of-fit test for a probability distribution varyingtemporally, and the test possible range of the goodness-of-fit test canbe widened.

According to the present invention, the widths of the counting sectionsare changed, so that the test possible range of the goodness-of-fit testcan be widened.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be described in detail basedon the following figures, wherein:

FIG. 1 is a view showing a graph in which a Poisson distribution with anarrival rate of 5 and observed data are overlapped and are indicated;

FIG. 2 is a block diagram showing a structure of a distributiongoodness-of-fit test device 1 of an embodiment;

FIG. 3 is a view for explaining the principle of the invention;

FIG. 4 is a view showing a change in distribution shapes due to a changein arrival rates;

FIG. 5 is a flowchart showing an operation procedure of a distributiongoodness-of-fit test device;

FIG. 6 is a view showing the shape of a Poisson distribution at anexpected frequency λ, and observed values of observation estimated tofollow the Poisson distribution and their frequencies;

FIG. 7 is a view showing the shape of a Poisson distribution at anexpected frequency λ, and observed values of observation estimated tofollow the Poisson distribution and their frequencies;

FIG. 8 is a view showing the shape of a Poisson distribution at anexpected frequency λ, and observed values of observation estimated tofollow the Poisson distribution and their frequencies;

FIG. 9 is a view showing the shape of a Poisson distribution at anexpected frequency λ, and observed values of observation estimated tofollow the Poisson distribution and their frequencies;

FIG. 10 is a view showing the shape of a Poisson distribution at anexpected frequency λ, and observed values of observation estimated tofollow the Poisson distribution and their frequencies;

FIG. 11 is a view showing the shape of a Poisson distribution at anexpected frequency λ, and observed values of observation estimated tofollow the Poisson distribution and their frequencies;

FIG. 12 is a view showing the shape of a Poisson distribution at anexpected frequency λ, and observed values of observation estimated tofollow the Poisson distribution and their frequencies;

FIG. 13 is a view showing the shapes of the Poisson distributions at therespective expected frequencies collectively;

FIG. 14 is a view showing the shapes of the Poisson distributions at therespective expected frequencies collectively;

FIG. 15 is a view showing a Poisson distribution with an arrival rate of1;

FIG. 16 is a flowchart showing a procedure of a section data countingprocessing;

FIG. 17 is a block diagram showing a structure of a printer apparatus10;

FIG. 18 is flowchart showing a procedure of a document receivingprocessing;

FIG. 19 is a flowchart showing a procedure of a print processing;

FIG. 20 is a flowchart showing a procedure of a steady data calculationprocessing;

FIG. 21 is a flow chart showing a procedure of a page number countingprocessing;

FIG. 22 is a flowchart showing a procedure of an arrival time recordingprocessing;

FIG. 23 is a flowchart showing a procedure of a supply judgmentprocessing; and

FIG. 24 is a view showing a whole structure of a computer for executinga program.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the present invention will be described with reference tothe accompanying drawings.

Embodiment 1

First, the structure of a distribution goodness-of-fit test device 1will be described with reference to FIG. 2. As shown in FIG. 2, thedistribution goodness-of-fit test device 1 includes a data holding part2, a distribution estimation part 3, a counting section determinationpart 4, a data counting part 5, and a goodness-of-fit test part 6.

Data obtained by observation is recorded in the data holding part 2. Thedistribution estimation part 3 reads out the observed data from the dataholding part 2 and estimates an arrival rate λ. The distributionestimation part 3 obtains an average arrival rate of the observed datafor each time zone, and obtains a posting rate assumed to beproportional to the number of participants in a mailing list.Incidentally, the arrival rate λ is the average value of the number ofarrivals per unit time, and corresponds to the average value of aPoisson distribution.

When a discrete distribution is estimated by the distribution estimationpart 3, as shown in FIG. 3, the counting section determination part 4determines counting sections so that widths of the respective countingsections have equal probabilities on the estimated discretedistribution.

In the case of a inhomogeneous Poisson process, the distribution shapeis changed by the arrival rate λ. That is, the discrete distributionvaries temporally. Thus, a chi-square goodness-of-fit test can not beapplied as it is. FIG. 4 shows the shapes of Poisson distributions inthe case where the arrival rate λ is 1, 2, 5 and 10. As shown in FIG. 4,when the arrival rate is changed, the shape of the Poisson distributionis also significantly changed. Then, as shown in FIG. 3, the countingsections are determined so that the widths of the respective countingsections have equal probabilities on the estimated discretedistribution, and the measured data is classified for each countingsection. By this, the plural estimated discrete distributions can beconverted into a discrete uniform distribution as shown in FIG. 3, andthe chi-square goodness-of-fit test can be performed as thegoodness-of-fit test to the uniform distribution. Accordingly, even ifthe distribution shape varies by the variation of the arrival rate λ,the chi-square goodness-of-fit test can be applied. Incidentally, in thecase where the chi-square goodness-of-fit test is used, in order toraise the accuracy of approximation to the chi-square distribution, itis appropriate that the number m of the counting sections is made (thenumber of data/10) or less.

In the data counting part 5, the numbers of data in the respectivecounting sections determined by the counting section determination part4 are counted. In the goodness-of-fit test part 6, a test method such asthe chi-square goodness-of-fit test is used to test the counted data.The number of sections is made m, and a test statistic obtained from amathematical expression set forth below is compared with a chi-squarevalue χ²(m−1, α) with a degree of freedom of m−1. When the teststatistic is larger than the chi-square value, it is judged that thedata does not follow the estimated discrete distribution. Incidentally,X_(i) denotes the number of data in an ith section, n denotes the totalnumber of data, m denotes the number of sections, and α denotes asignificance level. As the significance level α, 0.05 (5%) or 0.01 (1%)is used. $\begin{matrix}{\chi_{2}^{0} = {\sum\limits_{i = 1}^{m}\quad\frac{( {x_{i} - \frac{n}{m}} )^{2}}{\frac{n}{m}}}} & (3)\end{matrix}$

Next, the processing procedure of the distribution goodness-of-fit testdevice will be described with reference to a flowchart shown in FIG. 5.In a distribution goodness-of-fit test estimation processing, first, aparameter (arrival rate) of a discrete distribution is estimated from anobserved value or the like (step S1) In the case of a Poisson process(homogeneous Poisson process) in which the arrival rate is constant,estimation of the arrival rate λ is performed. In the case of aninhomogeneous Poisson process, estimation of the varying arrival rate λis performed. For example, an average arrival rate of observed data ofthe inhomogeneous Poisson process for each time zone is obtained, or aposting rate assumed to be proportional to the number of participants ina mailing list is obtained.

Next, in a counting section determination processing, counting sectionsare determined so that the widths of the respective counting sectionshave equal probabilities on the estimated distribution (step S2). First,in order to determine the number of the counting sections, a naturalnumber not larger than (observed frequency÷10) and not less than 2 isselected (step S3). Here, when the number of observed data is 19 orless, the number of sections becomes 1 (step S3/NO), and therefore, thetest is not performed and is ended. In this case, it may be displayed ornotified that the test is impossible.

When the number m of the counting sections is determined, the widths ofthe sections are determined so that expected values of the respectivesections are equal, that is, they have equal probabilities. When theinhomogeneous Poisson process is cited as an example, the probabilitythat the number of arrivals in a time width of from t to t+s becomes iis madeP[N(t+s)−N(t)=i]  (4),and as a section in which a cumulative probability from i=0 does notexceed 1/m, and a next section in which it does not exceed 1/m or asection in which a cumulative probability does not exceed 2/m, k₁, k₂, .. . , k_(m−1) of following expressions (5), (6), (7), (8), (9) or (10),(11), (12), (13) are determined. $\begin{matrix}{p_{1} = {{\sum\limits_{i = 0}^{k_{1}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{1}{m}}} & (5) \\{p_{2} = {{\sum\limits_{i = {k_{1} + 1}}^{k_{2}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{1}{m}}} & (6) \\{p_{j} = {{\sum\limits_{i = {k_{j - 1} + 1}}^{k_{j}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{1}{m}}} & (7) \\{p_{m - 1} = {{\sum\limits_{i = {k_{m - 2} + 1}}^{k_{m - 1}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{1}{m}}} & (8) \\{p_{m} = {{1 - {\sum\limits_{i = 1}^{m - 1}\quad P_{i}}} \cong \frac{1}{m}}} & (9) \\{p_{1} = {{\sum\limits_{i = 0}^{k_{1}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{1}{m}}} & (10) \\{{{\sum\limits_{i = 0}^{k_{2}}\quad p_{i}} = {{\sum\limits_{i = 0}^{k_{2}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{2}{m}}}\quad} & (11) \\{{\sum\limits_{i = 0}^{k_{j}}\quad p_{i}} = {{\sum\limits_{i = 0}^{k_{j}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{j}{m}}} & (12) \\{{\sum\limits_{i = 0}^{k_{m - 1}}\quad p_{i}} = {{\sum\limits_{i = 0}^{k_{m - 1}}\quad{P\lbrack {{{N( {t + s} )} - {N(t)}} = i} \rbrack}} \leq \frac{m - 1}{m}}} & (13)\end{matrix}$

In a section data counting processing, the numbers of data in therespective determined counting sections are counted (step S4). In thechi-square test processing, the chi-square goodness-of-fit test isperformed (sep S5). In the case of the chi-square goodness-of-fittest,the number of the sections is made m, and the test statistic is comparedwith the chi-square value λ²(m−1, α) (step S5). When the test statisticis larger than the chi-square value, it is judged that the estimateddistribution is not suitable.

As stated above, in this embodiment, even in the case where a sufficientnumber of data are not obtained as compared with the division numberaccording to the division by a conventional discrete value, the divisionnumber can be decreased, and therefore, the test with high accuracybecomes possible. Besides, a discrete distribution varying temporally isconverted into a uniform distribution of 1/m (m is the division number),so that the test can be performed also for the discrete distributionvarying temporally. Thus, it is possible to easily statistically testwhether the data follows the distribution function having a fluctuatingparameter and estimated using the data after observation. By this, inthe case where there is periodicity in variation for each time zonealthough there is a temporal change, it is possible to make a diagnosisof whether a large change occurs from a steady state.

Incidentally, when the foregoing method is not used, as long as anobservation time interval is not changed at the time of counting, thegoodness-of-fittest to a distribution function having a fluctuatingparameter is difficult to perform. Since it is almost impossible tochange the time interval while estimating the parameter at the time ofobservation, a method in which cost is high must be adopted, forexample, a large amount of storage capacity is required to recordcontinuously in time, or counting is again performed after the change ofthe parameter is estimated.

SPECIFIC EXAMPLE 1

Next, a specific example of a test using the foregoing distributiongoodness-of-fit test method will be described. First, a description willbe given to a case where statistical data which is shown in Table 1 andin which an arrival rate follows a specific Poisson distribution processis subjected to a goodness-of-fit test in accordance with the invention.Table 1 shows the number of calls received by a telephone line on aweekday. Table 3 set forth below shows the number x of arrivals on eachday, a cumulative Poisson probability of the arrival number, and thenumber of the counting section classified in accordance with theinvention. Table 4 shows a table in which the data are counted inaccordance with the classification shown in Table 3. TABLE 3 xcumulative Poisson probability classification 1 6 0.762183463 3 2 30.265025915 1 3 3 0.265025915 1 4 5 0.615960655 2 5 3 0.265025915 1 6 50.615960655 2 7 5 0.615960655 2 8 4 0.440493285 2 9 5 0.615960655 2 10 60.762183463 3 11 7 0.866628326 3 12 1 0.040427682 1 13 8 0.931906365 314 5 0.615960655 2 15 5 0.615960655 2 16 5 0.615960655 2 17 100.986304731 3 18 8 0.931906365 3 19 3 0.265025915 1 20 5 0.615960655 221 4 0.440493285 2 22 7 0.866628326 3 23 8 0.931906365 3 24 30.265025915 1 25 6 0.762183463 3 26 3 0.265025915 1 27 3 0.265025915 128 3 0.265025915 1 29 9 0.968171943 3 30 2 0.124652019 1

TABLE 4 classification observed frequency expected frequency 1 10 10 210 10 3 10 10

First, the arrival rate λ is obtained by Σ (arrival number×observedfrequency)/(total observed frequency) as described above and is made 5.Since the observed frequency is 30 in total, the number ofclassifications, that is, the number of counting sections is required tobe made 30/10 or less from the accuracy of approximation to thechi-square distribution, and it is made 3.

The cumulative Poisson probability expresses a cumulative probability inwhich the number of messages is from 0 to k in the case where thePoisson process is presupposed, and the arrival number is made k. Thecumulative Poisson probability is obtained from 0 to the value of each xshown in Table 3. For example, the cumulative Poisson probability inwhich the value of x is up to 3 becomes${{\frac{1}{{\mathbb{e}}^{5}} \cdot \frac{5^{0}}{0!}} + {\frac{1}{{\mathbb{e}}^{5}} \cdot \frac{5^{1}}{1!}} + {\frac{1}{{\mathbb{e}}^{5}} \cdot \frac{5^{2}}{2!}} + {\frac{1}{{\mathbb{e}}^{5}} \cdot \frac{5^{3}}{3!}}} = 0.26502591$

With respect to the determination of the classified counting section,even if a separation between the counting sections in the estimateddistribution is not directly calculated, when the cumulative Poissonprobability is calculated, an integer portion of a value obtained byadding 1 to cumulative Poisson probability×number of sections becomesthe number of the section.

As shown in Table 4, the result of the judgment becomes χ²(3−1,0.05)=5.991476357>0, and similarly to the conventional method, theresult is such that “it can not be said that the data does not followthe Poisson distribution”, and even in the case where there is novariation in arrival rate, a contradiction does not occur.

SPECIFIC EXAMPLE 2

As a specific example 2, a description will be given to an example inwhich it is tested whether the number of new messages of a mailing listfor each week follows a inhomogeneous Poisson process. In the mailinglist, it is conceivable that as the number of persons becomes large, thenumber of messages increases. Thus, a check is made for the case inwhich it is assumed that the posting rate (arrival rate), that is, theaverage number of messages varies in proportion to the number ofparticipants. However, since a response is significantly influenced by aformer message, a new message is made an object. The number ofregistrants in each week and the number of messages are known, and it istested whether the occurrence rate (arrival rate) of messages matchesthe inhomogeneous Poisson process in proportion to the number ofregistrants. Table 5 is a table in which the number of messages, thenumber of registrants, the expected value and the cumulative Poissonprobability are collected in each week. TABLE 5 number of number ofcumulative messages registrants expected value probabilityclassification 1 4 30 2.115671642 0.936307752 3 2 2 30 2.1156716420.645401408 2 3 1 30 2.115671642 0.375601365 2 4 0 30 2.1156716420.120552294 1 5 3 60 4.231343284 0.389625748 2 6 2 60 4.2313432840.206126414 1 7 7 60 4.231343284 0.933884263 3 8 1 60 4.2313432840.076026357 1 9 6 80 5.641791045 0.663628706 2 10 9 80 5.6417910450.938534165 3 11 5 80 5.641791045 0.504784723 2 12 4 80 5.6417910450.335855408 2 13 5 100 7.052238806 0.294086357 1 14 5 100 7.0522388060.294086357 1 15 1 100 7.052238806 0.006968964 1 16 4 100 7.0522388060.168279184 1 17 4 100 7.052238806 0.168279184 1 18 8 100 7.0522388060.722255536 1 19 9 110 7.757462687 0.746224814 3 20 9 110 7.7574626870.746224814 3 21 7 110 7.757462687 0.487296495 3 22 11 110 7.7574626870.904784248 2 23 10 110 7.757462687 0.83920934 3 24 11 120 8.4626865670.851827371 3 25 8 120 8.462686567 0.528241441 3 26 11 120 8.4626865670.851827371 2 27 10 120 8.462686567 0.764767283 3 28 14 120 8.4626865670.973458881 3 29 10 120 8.462686567 0.767467283 3 30 8 130 9.1679104480.433744878 2

In the case where it is assumed that the occurrence rate (arrival rate)of message is proportional to the number of registrants, the expectedvalue indicates an expected value of the number of messages calculatedfrom the number of registrants. A computation expression is as follows.expected value=(total number of messages)×(number of registrants at thattime)/(total number of registrants)

In the case where the Poisson probability is presupposed, the number ofmessages is made k, and the cumulative Poisson probability indicates acumulative probability in which the number of messages is from 0 to k.In this example, since the expected value, that is, the arrival rate λvaries, the Poisson distribution also varies in accordance with this.Thus, also with respect to the cumulative probability, the Poissondistribution corresponding to the expected value (arrival rate) isobtained, and the cumulative probability must be obtained from thisdistribution.

With respect to the calculation of the counting section to beclassified, it is obtained by multiplying the value of the cumulativePoisson probability by the number of sections and adding 1. The divisionnumber is determined on the basis of the total number of data. When theexpected frequency becomes 10 or more, since it may be approximated tothe chi-square distribution practically, the maximum value of thedivision number is made (number of data/10). In this example, the totalnumber of data is 30, and from the division number≦3.0, the divisionnumber is made 3.

When the division number is made 3 and the chi-square test is performed,Table 6 below is obtained. TABLE 6 observed expected classificationfrequency frequency (Xi − n/m)² ÷ (n/m) 1  8 10 0.4 2  9 10 0.1 3 13 100.9 total 30 30 1.4

Since it is premised that the expected frequency has equal probability,it is the total number of cases÷the number of classifications (number ofcounting sections) and is 10. Here, X_(i) is made each observedfrequency, the sum of(X_(i)−n/m)²/(n/m)is made a test statistic, and the goodness of fit is tested by making acomparison with a chi-square value. In this example, since it becomesχ²(3−1, 0.05)=5.991476357>1.4, it is possible to judge that the datamatches the inhomogeneous Poisson process. In this way, it is possibleto test whether or not the data already counted at constant timeintervals match the inhomogeneous Poisson process.

Incidentally, FIGS. 6 to 12 show the shapes of Poisson distributions atrespective expected frequencies λ, and observed values of observationestimated to follow the Poisson distributions and their frequencies.FIGS. 13 and 14 collectively show the shapes of the Poissondistributions at the respective expected frequencies in FIGS. 6 to 12.

SPECIFIC EXAMPLE 3

As a specific example 3, a description will be given to an example inwhich it is tested whether the degree of use of a printer follows theinhomogeneous Poisson process based on the number of attended persons.Also with respect to the degree of use of the printer, it is conceivablethat as the number of persons becomes large, the number of outputsincreases. Thus, it is assumed that the average number of times ofoutput varies in proportion to the number of attended persons. Here, theaverage number of attended persons and the number of outputs for eachworking time on a weekday are known, and it is tested whether or not theoccurrence rate of the output matches the inhomogeneous Poisson processin proportion to the number of attended persons. Table 7 is a table inwhich the number of outputs, the number of attended persons, theexpected value, and the cumulative Poisson probability are collected ineach week. TABLE 7 number of number of attended cumulative outputspersons expected value probability classification 1 11 80 7.2831541220.932809438 3 2 4 50 4.551971326 0.522268318 2 3 0 30 2.7311827960.065142194 1 4 3 25 2.275985663 0.804211299 3 5 2 30 2.7311827960.486016955 2 6 2 35 3.186379928 0.382753538 2 7 3 40 3.6415770610.506413382 2 8 5 60 5.462365591 0.535380044 2 9 6 80 7.2831541220.408427795 2 10 1 35 3.186379928 0.172986184 1 11 6 80 7.2831541220.408427795 2 12 7 50 4.551971326 0.909071299 3 13 2 30 2.7311827960.486016955 2 14 1 25 2.275985663 0.336429429 2 15 4 30 2.7311827960.858232777 3 16 2 35 3.186379928 0.382753538 2 17 1 40 3.6415770610.121660261 1 18 7 60 5.462365591 0.81410721 3 19 13 80 7.2831541220.982658332 3 20 3 35 3.186379928 0.605553033 2 21 9 80 7.2831541220.800662605 3 22 3 50 4.551971326 0.333603553 2 23 3 30 2.7311827960.70720591 3 24 3 25 2.275985663 0.804211299 3 25 1 30 2.7311827960.243057434 1 26 3 35 3.186379928 0.605553033 2 27 4 40 3.6415770610.6984703 3 28 6 60 5.462365591 0.691938616 3 29 10 80 7.2831541220.880168316 3 30 2 35 3.186379928 0.382753538 2

The expected value shown in Table 7 indicates the expected value of thenumber of outputs calculated from the number of attended persons in thecase where it is assumed that the occurrence rate (arrival rate) of theoutput is proportional to the number of attended persons. A calculationmethod is (total number of outputs)×(number of attended persons at thattime point)/(total number of attended persons).

As described above, in the case where the Poisson process ispresupposed, the number of outputs is made k, and the cumulative Poissonprobability indicates the cumulative probability in which the number ofoutputs is from 0 to k. With respect to the classification, the countingsection to be classified by multiplying the cumulative Poissonprobability by the number of sections and adding 1 is calculated. Thedivision number is determined on the basis of the total number of data.When the expected frequency becomes 10 or more, it may be approximatedto the chi-square distribution practically, the maximum value of thedivision number is made (number of data/10). In this example, the totalnumber of data is 30, and from the division number≦3.0, the divisionnumber is made 3.

When the division number is made 3, and the chi-square goodness-of-fittest is performed, Table 8 shown below is obtained. TABLE 8 observedexpected classification frequency frequency (Xi − n/m)² ÷ (n/m) 1 4 103.6 2 14 10 1.6 3 12 10 0.4 total 30 30 5.6

Since it is premised that the expected frequency has equal probability,it is the total number of cases÷the number of classifications (number ofcounting sections). Here, X_(i) is made each observed frequency, thetotal of(X_(i)−n/m)²/(n/m)is made a test statistic, and the goodness of fit is tested by making acomparison with a chi-square value. In this example, because of χ²(3−1,0.05)=5.991476357>5.6, it is understood that the data is judged to matchthe inhomogeneous Poisson process.

Embodiment 2

The details of this embodiment will be described with reference to theaccompanying drawings. In this embodiment, in the case where the Poissonprobability of a discrete value extends over plural counting sections,the rates at which the Poisson probability of the discrete value isincluded in the respective counting sections are obtained, and they areclassified into the respective counting sections.

For example, as shown in FIG. 15, in the case where the arrival rate is1, the Poisson probability at which the discrete value becomes 0 isabout 0.368. In the case where the division number is made 4, sincedividing into quarters is performed, the width of each counting sectionon the Poisson probability becomes 0.25. In this case, since the Poissonprobability of a discrete value of 0 is about 0.368, it is included inthe counting section 1 and the counting section 2. Then, the ratios atwhich the Poisson probability of 0.368 for the discrete value isincluded in the respective counting sections are obtained. The ratio atwhich it is included in the counting section 1 is (0.25/0.368), and theratio at which it is included in the counting section 2 is(0.368−0.25)/0.368.

In this way, in the case where an error becomes large when counting isperformed in either one of counting sections, counting can be performedwhile division into the plural counting sections is performed, so thatthe error can be made small.

A procedure of the division of the Poisson probability of a discretevalue will be described with reference to a flowchart shown in FIG. 16.First, initial setting is performed (step S10). A cumulative Poissonprobability at which the discrete value becomes k or less is made P_(k),and P₀ is made P₀=0. A division number is made m, a storage area forcounting is initialized to establish X₁=X₂= . . . =X_(m)=0. Next, acoefficient i is made k−1 (step S2). The character k denotes a discretevalue of data to be counted, and i denotes a value smaller than k by 1.

Next, h=int(mP_(i+1))+1 is calculated. The function int(x) indicates afunction to discard digits to the right of the decimal point of x. Thecharacter h takes a value of 1 to m, and is a coefficient indicating acounting section. The counting section in which the Poisson probabilityof the discrete value k is included is obtained by multiplying thecumulative probability at which the discrete value becomes k or less bythe division number and by adding 1.

Next, int(mP_(i)) is compared with int(mP_(i+1)), and it is judgedwhether or not these are equal (step S14). When the counting section isnot changed even if the discrete value is increased by 1 (step S14/YES),the whole Poisson probability of the discrete value k is included in thecounting section of h. At this time, since the whole Poisson probabilityof the discrete value k is included in one counting section, Xh is setto 1 (step S15).

In the case where int(mP_(i)) is not equal to int(mP_(i+1)) (stepS14/NO), the Poisson probability of the discrete value k extends overplural counting sections. In this case, the rates at which the Poissonprobability of the discrete value k is included in the respectivecounting sections are obtained, and they must be classified into therespective counting sections.

First, the rate at which the Poisson probability of the discrete value kis included in the counting section h is obtained. This value becomes(h/m−P_(i))/(P_(i+1)−P_(i)) (step S17). The portion (h/m−P_(i)) denotesthe width of the counting section from the cumulative probability of k−1to the threshold (h/m) between the respective counting sections, andP_(i+1)−P_(i) denotes the Poisson probability of the discrete value k.In the example of λ=1 and m=4 shown in FIG. 15, when the discrete valueis made k=0, it corresponds to the portion of 0.25/(0.368−0).

Next, a variable j (j is an integer not smaller than 1) with an initialvalue of 1 is used (step S18), an integer value of 1 or more is added tothe counting section h, and it is judged whether or not the Poissonprobability of the discrete value k extends over two or more countingsections. That is, it is judged whether or not the threshold of a nextcounting section is smaller than the cumulative Poisson probability of k(step S19). A judgment expression is (h+j)/m≦P_(i+1).

An integer of 1 or more is added to the counting section h, and when thethreshold of the next counting section is smaller than the cumulativePoisson probability of k (step S19/YES) since it extends across the nextthreshold, the width from the threshold to the threshold is divided bythe width of the Poisson probability of the discrete value k, and therate is obtained. This rate becomes (1/m) (P_(i+1)−P_(i)) (step S20).The value j is added with 1 (step S21), and this processing is repeateduntil the threshold becomes larger than the cumulative Poissonprobability of k.

Besides, an integer of 1 or more is added to the counting section h, andwhen the threshold of a next counting section is larger than thecumulative Poisson probability of k (step S19/NO), the width from thethreshold across which it finally extends to the cumulative Poissonprobability of k is obtained, the width is divided by a difference(width) of the cumulative Poisson probability of from k−1 to k, and therate is obtained. This rate becomes{P_(i+1)−(h+j−1)}/m/(P_(i+1)−P_(i))(step S22). This is performed for all observed data, and when countingof all the data is ended (step S16/YES), the processing is ended.

SPECIFIC EXAMPLE 4

Next, a description will be given to an example in which division intoplural counting sections is made and counting is performed according tothe foregoing section data counting processing. Table 9 shows dataindicating the relation between the degree of use of the printer and thenumber of attended person listed in example 3, and Table 10 indicatescounting values in respective divided counting sections. TABLE 9 numbernumber of cumulative of attended expected probability cumulative outputpersons value (k − 1) probability (k) classification 1 classification 2classification 3 1 11 80 7.282154122 0.880168316 0.932809438 0 0 1 2 450 4.531971326 0.333603553 0.522268318 0 1 0 3 0 30 2.731182796 00.065142194 1 0 0 4 3 25 2.275985663 0.60241681 0.804211299 00.318392523 0.681607477 5 2 30 2.731182796 0.243057434 0.4860169550.371567633 0.628432347 0 6 2 35 3.186379928 0.172986184 0.3827535380.76440469 0.23559531 0 7 3 40 3.641577061 0.295453325 0.5064133820.179560487 0.820439513 0 8 5 60 5.462365591 0.363412164 0.535380044 0 10 9 6 80 7.283454122 0.266014393 0.408427795 0.473700881 0.527299119 010 1 35 3.186379928 0.041321186 0.172986184 1 0 0 11 6 80 7.2831541220.266014393 0.408427795 0.472700881 0.527299119 0 12 7 50 4.5519713260.824334889 0.909071299 0 0 1 13 2 30 2.731182796 0.2430574340.486016955 0.371567653 0.628432347 0 14 1 25 2.275985663 0.1026956350.336429429 0.986753754 0.013246246 0 15 4 30 2.731182796 0.707205910.858232777 0 0 1 16 2 35 3.186379928 0.172986184 0.382753538 0.764404690.23559531 0 17 1 40 3.641577061 0.026210975 0.121660261 1 0 0 18 7 605.462365591 0.691938616 0.81410721 0 0 1 19 13 80 7.2831541220.964758888 0.982658331 0 0 1 20 3 35 3.186379928 0.3827535380.605553033 0 1 0 21 9 80 7.283154122 0.69149876 0.800662605 0 0 1 22 350 4.551971326 0.167816247 0.333603533 0.998370085 0.001629915 0 23 3 302.731182796 0.486016955 0.70720591 0 0.816721211 0.183278789 24 3 252.275985663 0.60241681 0.804211299 0 0318392523 0.681607477 25 1 302.731182796 0.065142194 0.243057434 1 0 0 26 3 35 3.1863799280.382753538 0.605553033 0 1 0 27 4 40 3.641577061 0.506413382 0.69847030 0.834405169 0.165594831 28 6 60 5.462365591 0.535380044 0.691938616 00.83857831 0.16142169 29 10 80 7.283154122 0.800662605 0.880168316 0 0 130 2 35 3.186379928 0.172986184 0.382753538 0.76440469 0.23559531 0

TABLE 10 observed expected classification frequency frequency (Xi −n/m)² ÷ (n/m) 1 10.14643546 10 0.002144334 2 10.98005427 10 0.0960506383 8.873510264 10 0.126897913 total 30 30 0.225092885

The classification 1, the classification 2 and the classification 3shown in Table 9 indicate numerical values counted according to therates at which the Poisson probability of the discrete value is includedin the respective counting sections 1, 2 and 3. As a result, as shown inTable 10, because of χ²(3−1, 0.05)=5.991476357>0.225092885, it ispossible to judge that the data well matches the inhomogeneous Poissonprocess.

SPECIFIC EXAMPLE 5

As a specific example 5, a description will be given to an example inwhich it is tested whether or not the degree of use of a smoking room ofa main office matches the inhomogeneous Poisson process. Also withrespect to the degree of use of the smoking room of the main office of acertain company, it is conceivable that as the number of persons becomeslarge, the number of users increase. Thus, a test is performed on theassumption that the average number of users per time varies inproportion to the number of persons staying in their room. However, inthe case of the smoking room, since it is conceivable that nonsmokers donot use, an expected number of persons is obtained by multiplying thenumber of persons staying in the room by a smoker rate. Here, an averageexpected number of persons and the number of users for each working timeon a weekday are known, and it is tested whether the use rate matchesthe inhomogeneous Poisson process in proportion to the expected numberof persons.

Table 11 shows the number of users on each day, the expected number ofpersons, an expected value, a cumulative Poisson probability up to adiscrete value k−1, a cumulative Poisson probability up to a discretenumber k, and rates at which the cumulative Poisson probability of thediscrete number k is included in counting sections 1, 2 and 3. Table 12is a table in which data are re-edited in the counting sections 1, 2 and3. TABLE 11 expected cumulative cumulative number number of expectedprobability probability of users persons value (k − 1) (k)classification 1 classification 2 classification 3 1 15 20 13.695090440.602717891 0.699160922 0 0.663073758 0.336926242 2 11 16 10.956072350.465146045 0.584510607 0 1 0 3 1 8 5.478036176 0.004177526 0.0270621621 0 0 4 0 1.2 0.821705426 0 0.439681171 0.758125104 0.241874896 0 5 33.2 2.19121447 0.625070462 0.821077226 0 0.212218209 0.787781791 6 5 6.84.656330749 0.502710181 0.676018381 0 0.946039979 0.053960021 7 7 106.84954522 0.47266535 0.621417318 0 1 0 8 19 15.2 10.408268730.989407738 0.994714665 0 0 1 9 14 18 12.3255814 0.646748473 0.7418009550 0.20954943 0.79045057 10 8 4.8 3.286821705 0.980630892 0.993256145 0 01 11 19 20 13.69509044 0.896809564 0.935279614 0 0 1 12 11 1610.95607235 0.465146045 0.584510607 0 1 0 13 6 8 5.47036176 0.5326870120.689483379 0 0.854481882 0.145518118 14 1 12 0.821705426 0.4396811710.800969575 0 0.628266762 0.371733238 15 1 3.2 2.19121447 0.1117809120.356716862 0.904532066 0.095467934 0 16 2 6.8 4.656330749 0.0537422750.156742672 1 0 0 17 7 10 5.94754522 0.47266535 0.621417618 0 1 0 18 1215.2 10.40826873 0.64982321 0.751370932 0 0.168324399 0.831675601 19 918 12.3255814 0.134830501 0.21510391 1 0 0 20 1 4.8 3.2868217050.037372441 0.160208993 1 0 0 21 10 20 13.69509044 0.0246391180.196780392 1 0 0 22 10 16 10.95607235 0.345296886 0.465143045 0 1 0 235 8 5.478036176 0.360950613 0.532687012 0 1 0 24 0 1.2 0.821705426 00.139681171 0.758125104 0.241874896 0 25 3 3.2 2.19121447 0.6250704620.821077226 0 0.212218209 0.787781791 26 3 6.8 4.656330749 0.1567426720.316610643 1 0 0 27 7 10 6.8754522 0.47266535 0.621417318 0 1 0 28 1315.2 10.40826873 0.751370932 0.832913991 0 0 1 29 7 18 12.32558140.038171486 0.076215766 1 0 0 30 2 4.8 3.286821705 0.1602039930.362079916 0.857599193 0.142400807 0

TABLE 12 observed expected classification frequency frequency (Xi −n/m)² ÷ (n/m) 1 10.27838147 10 0.00775 2 11.61579116 10 0.261078 38.105827372 10 0.358789 total 30 30 0.627617

The expected value indicates the expected value of users calculated fromthe expected number of persons in the case where the use rate (arrivalrate) is proportional to the expected number of persons. The calculationmethod is (total number of users)÷(total expected number ofpersons)×(expected number of persons at that time point). Theclassification 1, the classification 2 and the classification 3respectively indicate numerical values counted in the respectivecounting sections 1, 2 and 3.

As a result, because of χ²(3−1, 0.05)=5.991476357>0.627617, it isunderstood that the degree of use well matches the inhomogeneous Poissonprocess.

SPECIFIC EXAMPLE 6

As a specific example 6, a description will be given to an example inwhich it is tested whether the degree of use of a smoking room of abusiness establishment matches the inhomogeneous Poisson process. Alsowith respect to the degree of use of the smoking room of anotherbusiness establishment of a certain company, it is conceivable that asthe number of persons becomes large, the number of users increases.Thus, a test is performed on the assumption that the average number ofusers per time varies in proportion to the number persons staying intheir room.

Table 13 shows the number of users on each day, the expected number ofpersons, an expected value, a cumulative Poisson probability up to adiscrete value k−1, a cumulative Poisson probability up to a discretevalue k, and rates at which the cumulative Poisson probability of thediscrete value k is included in counting sections 1, 2 and 3. Table 14is a table in which data are re-edited in classifications 1, 2 and 3.TABLE 13 expected number cumulative cumulative number of expectedprobability probability of users persons value (k − 1) (k)classification 1 classification 2 classification 3 1 1 11 2.0523860780.128428098 0.392012139 0.777381037 0.22618963 0 2 4 10 1.8658055260.880490415 0.958643335 0 0 1 3 1 8 1.492644421 0.224777463 0.5602902880.323552074 0.676447926 0 4 2 7.5 1.399354144 0.592055707 0.833653797 00.30882264 0.69117736 5 1 6.5 1.585934697 0.204756319 0.5294864690.395950344 0.604049658 0 6 0 9.5 1.775515249 0 0.169905097 1 0 0 7 310.5 1.959095802 0.687746419 0.864428 0 0 1 8 3 10.8 2.0150699680.672597496 0.854393936 0 0 1 9 0 11.1 2.071044133 0 0.126054036 1 0 010 1 6 1.119483315 0.326448422 0.691901984 0.481763806 0.4817638060.036472387 11 2 11 2.052386078 0.392012139 0.662500247 0 1 0 12 4 101.865805526 0.880490415 0.958643335 0 0 1 13 0 8 1.492644421 00.224777463 1 0 0 14 1 7.5 1.399354144 0.246756281 0.5920557070.250730368 0.749269632 0 15 4 8.5 1.585934697 0.923113151 0.977084997 00 1 16 1 9.5 1.772515249 0.169908097 0.471064473 0.542663617 0.4573368380 17 2 10.5 1.959095802 0.417190614 0.687746419 0 0.9220872290.077912771 18 3 10.8 2.015069968 0.672597496 0.854393936 0 0 1 19 111.1 2.071044133 0.126054096 0.387117691 0.793979863 0.206020137 0 20 26 1.119483315 0.691901984 0.896461566 0 0 1 21 2 11 2.0523860780.392012139 0.662500247 0 1 0 22 3 10 1.865808826 0.7129425760.830490415 0 0 1 23 2 8 1.492644421 0.560290288 0.810890962 00.424824649 0.575175351 24 1 7.5 1.399354144 0.246756281 0.5920557070.250730368 0.749269632 0 25 2 8.5 1.585934697 0.529486469 0.786986875 00.532737793 0.467262207 26 2 9.5 1.772515249 0.471064473 0.737969266 00.732853807 0.267146193 27 1 10.5 1.959095802 0.140985842 0.4171906140.696394525 0.303605475 0 28 1 10.8 2.015069963 0.133311076 0.401942220.744598166 0.255401834 0 29 1 11.1 2.071044133 0.126054096 0.3871176910.793979863 0.206020137 0 30 1 6 1.119483315 0.326443422 0.6919019840.481763306 0.481783806 0.036472387

TABLE 14 observed expected classification frequency frequency (Xi −n/m)² ÷ (n/m) 1 9.533487838 10 0.021763 2 10.31489351 10 0.009916 310.15161866 10 0.002299 total 30 30 0.033977973

Similarly to the former example, in the case of the smoking room, sinceit is conceivable that nonsmokers do not use, the expected number ofpersons is obtained by multiplying the number of persons staying intheir room by a smoker rate. Here, the average expected number ofpersons for each working time on a weekday and the number of users areknown, and it is tested whether the use rate matches the inhomogeneousPoisson process in proportion to the expected number of persons.

In the case where it is assumed that the use rate (arrival rate) is inproportion to the expected number of persons, the expected valueindicates the expected value of the users calculated from the expectednumber of persons. The calculation method is (total number ofusers)÷(total expected number of persons)×(expected number of persons atthat time point).

The classification 1, the classification 2 and the classification 3indicate numerical values counted in the respective counting sections.

As a result, because of χ²(3−1, 0.05)=5.991476357>0.033977973, it isunderstood that the use rate well matches the inhomogeneous Poissonprocess.

SPECIFIC EXAMPLE 7

As a specific example 7, a description will be given to an example inwhich the observed data of specific example 5 and specific example 6 arecombined, and it is tested whether the degrees of use of the smokingrooms of the main office and the business establishment can be said tomatch the same inhomogeneous Poisson process. On the assumption that theaverage number of users per time similarly varies in proportion to thenumber of persons staying in their room, the description will be givento the example in which it is tested whether the use degrees of thesmoking rooms of the main office of a certain company and anotherbusiness establishment follow the same distribution. The number of usersand the expected number of persons are the data of the two formerexamples.

Table 15 shows the number of users of the main office on each day, theexpected number of persons, an expected value, a cumulative Poissonprobability up to a discrete value k−1, a cumulative Poisson probabilityup to a discrete value k, and rates at which the cumulative Poissonprobability of the discrete value k is included in counting sections 1,2 and 3. Table 16 shows the number of users of the businessestablishment on each day, the expected number of persons, the expectedvalue, a cumulative Poisson probability up to a discrete value k−1, acumulative Poisson probability up to a discrete value k, and rates atwhich the cumulative Poisson probability of the discrete value k isincluded in counting sections 1, 2 and 3. Table 17 is a table in whichthe data of the main office and the data of the business establishmentare re-edited in the counting sections 1, 2 and 3. TABLE 15 expectednumber cumulative number of expected probability cumulative of userspersons value (k − 1) probability (k) classification 1 classification 2classification 3 1 15 20 8.975012749 0.959337268 0.978445829 0 0 1 2 1116 7.180010199 0.888210407 0.93809669 0 0 1 3 1 8 3.590005099 0.027598190.126675831 1 0 0 4 0 1.2 0.538500765 0 0.583622584 0.5711453640.428854646 0 5 3 3.2 1.43600204 0.824731752 0.942131227 0 0 1 6 5 6.83.051504335 0.806535893 0.910800573 0 0 1 7 7 10 4.487506374 0.8326479230.914438968 0 0 1 8 19 15.2 6.821009689 0.999906268 0.999968743 0 0 1 914 18 8.077511474 0.963467648 0.981392647 0 0 1 10 8 4.8 2.154003060.996262312 0.999595767 0 0 1 11 19 20 8.975012749 0.9976449930.998977818 0 0 1 12 11 16 7.180010199 0.888210407 0.93809669 0 0 1 13 68 3.590005099 0.845491953 0.927549493 0 0 1 14 1 1.2 0.5385007650.583622584 0.897903792 0 0.26423496 0.73576504 15 1 3.2 1.436002040.237876882 0.579468571 0.279446058 0.720553942 0 16 2 6.8 3.0515043350.19158646 0.411750555 0.643823751 0.356176249 0 17 7 10 4.4875063740.332647923 0.914438968 0 0 1 18 12 15.2 6.821009689 0.9543267610.977421878 0 0 1 19 9 18 8.77511474 0.581729131 0.706961489 00.678239527 0.321760473 20 1 4.8 2.15400306 0.116018797 0.365923640.669589135 0.130410865 0 21 10 20 8.975012749 0.590700418 0.708947163 00.642438392 0.357581608 22 10 16 7.180010159 0.811783063 0.888210407 0 01 23 5 8 3.590005099 0708348627 0.845491953 0 0 1 24 0 1.2 0.538500765 00.583622584 0.571145364 0.428854636 0 25 3 3.2 1.43600204 0.8247317520.942131227 0 0 1 26 3 6.8 3.051504335 0.411750555 0.63569445 0 1 0 27 710 4.487506374 0.832647923 0.914438968 0 0 1 28 13 15.2 6.8210096890.977421878 0.989539725 0 0 1 29 7 18 8.077511474 0.3039991730.442194666 0.212265678 0.787734322 0 30 2 4.8 2.15400306 0.365923640.635071539 0 0 0

TABLE 16 expected number cumulative number of expected probabilitycumulative of users persons value (k − 1) probability (k) classification1 classification 2 classification 3 1 1 11 4.936257012 0.0071814280.042630803 1 0 0 2 4 10 4.487506374 0.344408241 0.534476596 0 1 0 3 1 83.590005099 0.02759819 0.126675831 1 0 0 4 2 7.5 3.365629781 0.1507899720.346418725 0.933120641 0.066879359 0 5 1 8.5 3.814380418 0.0220513730.106163698 1 0 0 6 0 9.5 4.263131056 0 0.014078154 1 0 0 7 3 10.54.711881693 0.151110879 0.307817614 1 0 0 8 3 10.8 4.8465058840.138189461 0.287236741 1 0 0 9 0 11.1 4.931132075 0 0.006866295 1 0 010 1 6 2.692503825 0.06771119 0.250023828 1 0 0 11 2 11 4.9362570120.042630803 0.130124416 1 0 0 12 4 10 4.487506374 0.3444082410.534476596 0 1 0 13 0 8 3.590005099 0 0.02759819 1 0 0 14 1 7.53.365629781 0.034540256 0.150789972 1 0 0 15 4 8.5 3.8143804180.47054725 0.685047609 0 1 0 16 1 9.5 4.263131056 0.0140781540.074095169 1 0 0 17 2 10.5 4.711881893 0.051337332 0.151110879 1 0 0 183 10.8 4.846506384 0.138169461 0.287236741 1 0 0 19 1 11.1 4.9811320750.006866285 0.041068157 1 0 0 20 2 6 2.692503825 0.250023828 0.4954625660.339430956 0.660569044 0 21 2 11 4.936257012 0.042630803 0.130124416 10 0 22 3 10 4.487506374 0.174988221 0.344408241 0.934630466 0.0653695340 23 2 8 3.590005099 0.126675831 0.304520451 1 0 0 24 1 7.5 3.3656297910.034540256 0.150789972 1 0 0 25 2 8.5 3.814380418 0.106163698 0.26658191 0 0 26 2 9.5 4.263131056 0.074095169 0.202025369 1 0 0 27 1 10.54.711881693 0.008987849 0.057337532 1 0 0 28 1 10.8 4.8465068840.007855771 0.045928818 1 0 0 29 1 11.1 4.981132075 0.0068662850.041068157 1 0 0 30 1 6 2.692503825 0.06771119 0.250023828 1 0 0

TABLE 17 observed expected classification frequency frequency (Xi −n/m)² ÷ (n/m) 1 30.35459741 20 5.360884 2 10.23031547 20 4.772337 319.41508712 20 0.017106 total 60 60 10.15032733

Here, the expected value indicates the expected value of userscalculated by adding the data of the main office and the data of thebusiness establishment and from the expected number of persons. Thecalculation method is the combination of the data of both the mainoffice and the business establishment and (total number of users)+(totalexpected number of persons)×(expected number of persons at that timepoint).

As is apparent from Table 15 and Table 16, since the average use rateper the number of persons is different from the case of only the mainoffice and the case of only the business establishment, a calculatedvalue after the expected value is different. As a result, because ofχ²(3−1 0.05)=5.991476357<10.15032733, it is understood that the use ratedoes not match the inhomogeneous Poisson process in proportion to theexpected number of persons.

In the two former examples, in each of the bases, since it can be saidthat the use rate matches the inhomogeneous Poisson process inproportion to the expected number of persons, it is possible to draw aconclusion that the use rate per the expected number of persons islargely different between the main office and the businessestablishment. In this way, it also becomes possible to perform thecomparison of varying distributions.

SPECIFIC EXAMPLE 8

As a specific example 8, a description will be given to an example inwhich a comparison is made between a past use rate of a parking lot anda present use rate. TABLE 18 9:00 10:00 11:00 12:00 13:00 14:00 15:0016:00 17:00 18:00 first day 2 6 2 11 6 13 7 13 1 2 second day 1 2 5 3 1016 17 15 0 0 third day 1 2 3 8 6 15 12 14 0 1 past average 4 6 10 6 1210 6 5 4 2

TABLE 19 arrival past number average classification 1 classification 2classification 3 1 2 4 1 0 0 2 6 6 0 1 0 3 2 10 1 0 0 4 11 6 0 0 1 5 612 1 0 0 6 13 10 0 0 1 7 7 6 0 0.404656279 0.595343721 8 13 5 0 0 1 9 14 1 0 0 10 2 2 0 0.952761133 0.047238867 11 1 4 1 0 0 12 2 6 1 0 0 13 510 1 0 0 14 3 6 1 0 0 15 10 12 0.807556779 0.192443221 0 16 16 10 0 0 117 17 6 0 0 1 18 15 5 0 0 1 19 0 4 1 0 0 20 0 2 1 0 0 21 1 4 1 0 0 22 26 1 0 0 23 3 10 1 0 0 24 8 6 0 0 1 25 6 12 1 0 0 26 15 10 0 0 1 27 12 60 0 1 28 14 5 0 0 1 29 0 4 1 0 0 30 1 2 0.722648929 0.277351071 0

TABLE 20 observed expected classification frequency frequency (Xi −n/m)² ÷ (n/m) 1 16.53020571 10 4.264359 2 2.827211704 10 5.144889 310.64258259 10 0.041291 total 30 30 9.450539092

Table 18 shows the comparison between the past average arrival rate ofthe pay-by-the-hour parking lot and the present arrival number of carswhile the arrival of cars follows the inhomogeneous Poisson process. Theway of obtaining counting sections is based on division in probabilityaccording to the assumed distribution parameter (past average) similarlyto the other examples, and a midway calculation result is partiallyomitted and is shown in Table 19.

As the result, as shown in Table 20, because of χ²(3−1,0.05)=5.991476357<9.450539092, it can be said that the use rate does notmatch the inhomogeneous Poisson process indicated by the past average.From this, it is understood that in terms of the arrival number of theparking lot for the last several days, the tendency of use is differentfrom the past average in the tendency.

SPECIFIC EXAMPLE 9

As a specific example 9, a description will be given to an example inwhich it is tested whether the use degree of a printer matches abinomial distribution. TABLE 21 number number of cumulative of attendedprobability cumulative outputs persons (k − 1) probability (k)classification 1 classification 2 classification 3 1 1 80 0.890577340.942028815 0 0 1 2 4 50 0.321303397 0.517316049 0.061373263 0.9386267370 3 0 30 0 0.057062577 1 0 0 4 3 25 0.599011084 0.811534793 00.318343694 0.631656306 5 2 30 0.228520319 0.47752618 0.4209258940.579074106 0 6 2 35 0.159522364 0.370853024 0.822459785 0.177540215 0 73 40 0.281877309 0.499955474 0.235952208 0.764047792 0 8 5 60 0.352377290.531618549 0 1 0 9 6 80 0.253240385 0.399645161 0.547065132 0.4529348680 10 1 35 0.035405979 0.159522364 1 0 0 11 6 80 0.253240385 0.3996451610.547065132 0.452934868 0 12 7 50 0.833607842 0.919024236 0 0 1 13 2 300.228520319 0.47752618 0.420925894 0.579074106 0 14 1 25 0.0919657570.322242789 1 0 0 15 4 30 0.710298221 0.867667173 0 0 1 16 2 350.159522364 0.370853024 0.822459785 0.177540215 0 17 1 40 0.0219685720.109981465 1 0 0 18 7 60 0.696182192 0.823331523 0 0 1 19 13 800.971660076 0.987183952 0 0 1 20 3 35 0.370853024 0.603683411 0 1 0 21 980 0.696334231 0.80985235 0 0 1 22 3 50 0.154746827 0.321303397 1 0 0 233 30 0.47752618 0.710298221 0 0.812556724 0.197443276 24 3 250.599011084 0.811534793 0 0.318343694 0.681656306 25 1 30 0.0570625770.228520319 1 0 0 26 3 35 0.370853024 0.603683411 0 1 0 27 4 400.499955474 0.70199595 0 0.8528137595 0.174862405 28 6 60 0.5316185490.696182192 0 0.8020643706 0.179356294 29 10 80 0.80985235 0.89057734 00 1 30 2 35 0.159522364 0.370853024 0.822459875 0.177540215 0

TABLE 22 expected classification observed frequency frequency (Xi −n/m)² ÷ (n/m) 1 10.70068688 10 0.049096 2 10.39433853 10 0.01555 38.904974587 10 0.119908 total 30 30 0.184554564

In the case where the number of attended persons is small, there is acase where it is more suitable to use the binomial distribution in whichthe probability at which one person prints out in a definite time is P.Here, a description will be given to an example in which the probabilityP is made (total number of outputs)/(total number of attendedpersons)=0.091039427, and the goodness of fit to the binomialdistribution in which the trial number n is made the number of attendedpersons is tested. Similarly to the Poisson distribution, the cumulativeprobability is used, and three classifications in probability areformed.

As a result, because of χ²(3−1, 0.05)=5.991476357>0.184554564, it isjudged that the degree of use well matches the binomial distribution inwhich the trial number n is made the number of attended persons.

As described above, in this example, since the arrival rate in each timeperiod can be estimated, in view of a parameter of a most crowded timezone, for example, it is possible to propose to the customer thespecifications for necessary speed and the like of a printer while thestatistic of the number of output sheets is taken into consideration, orin a design department of a printer, the specifications for a memorysize and the like necessary for a queue of a printer can be determinedwhile the statistic of the output file size is taken into consideration.

On the contrary, this example is integrated into a printer or the like,a statistic in each time period is counted, and it is possible to testwhether for example, the number of outputs on a certain a day isdifferent from a normal use state. When this is monitored on-line, itbecomes possible to make such a service such as to advance the supplytiming of consumable goods such as paper and toner, or to advance thetiming of maintenance. Similarly, even in a probability process varyingtemporally, such as the number of passengers getting on and off a trainand the number of guests coming to a store, an average arrival number ineach time zone is obtained and is made an estimated value of an arrivalrate, and it becomes possible to test whether the probability processcan be treated as a inhomogeneous Poisson process, to diagnose whetherdata on a certain day is data remote from a steady state, or to make acomparison of arrival degree of customers among plural stores.

Embodiment 3

Next, an embodiment in which the foregoing distribution goodness-of-fittest device 1 is provided in a printer apparatus 10 will be described.FIG. 17 shows a structure of the printer device 10. As shown in FIG. 17,the printer apparatus 10 includes a communication part 11, a documentstorage part 12, a control part 13, an image processing part 14, anoutput part 15, a notification part 16, and a consumable goods supplytiming judgment device 20 including the foregoing distributiongoodness-of-fit test device 1 and an arithmetic part 17.

The communication part 11 exchanges data with a PC as a client apparatusdirectly or through a network. A document received through thecommunication part 11 is stored in the document storage part 12. Thecontrol part 13 receives a print job through the communication part 11,and stores the print job into a print queue so that it can besequentially printed out. The print queue is a program which stores theprint job so that it can be sequentially printed out. The print queuesends the time when the storage of the document is completed to thedistribution goodness-of-fit test device 1, waits until the imageprocessing part 14 comes to have a state in which a processing is notbeing performed, and sends the document data stored in the documentstorage part 12 to the image processing part 14.

The image processing part 14 color-separates the document data toconvert it into a raster image for print output, and sends it to theoutput part 15. The output part 15 is a functional part for printing ona sheet or the like. The image processing part 14 sends information ofthe number of pages of the raster image sent to the output part 15 tothe distribution goodness-of-fit test device 1.

The consumable goods supply timing judgment device 20 includes theforegoing distribution goodness-of-fit test device 1 and the arithmeticpart 17. On the basis of the storage time of the document into thedocument storage part 12, the arithmetic part 17 obtains an averagevalue in a steady state at constant time intervals between the supply ofsheets and next supply and an average value of the number of pages perone time. The distribution estimation part 3 uses these data to estimatean arrival distribution, and stores it in the data holding part 2.

Besides, the arithmetic part 17 compares the obtained statistical datawith the time data of the document arrival from the final sheet supplyby the test method of the invention, and in the case where the arrivaldistribution is significantly different from the steady state and islarger than usual, or in the case where the average number of pages islarge, that is, in the case where consumption is high, a warning isissued from the notification part 16.

Besides, instead of the notification part 16, the network connection isprovided, and when notification is made through the network, a serviceof delivery can be performed before the sheet runs out, or it is alsopossible to perform the delivery service of the whole consumable goodsby estimating the number of pages in view of the use quantity of ink ortoner.

The detailed procedure of each processing will be described withreference to flowcharts. First, a procedure of a document receivingprocessing by the control part 13 will be described with reference to aflowchart shown in FIG. 18. When accepting a sending request for adocument, the control part 13 examines the free space of the queue (stepS21). In the case where the queue does not have the free space (stepS21/NO), it is notified through the communication part 11 that it isimpossible to receive the document (step S24). When it is possible toreceive the document (step S21/YES), a print job is added to the queue(step S22), and the time when the storage is completed is sent to thedistribution goodness-of-fit test device 1 (step S23).

Next, a procedure of a print processing by the image processing part 14will be described with reference to a flowchart shown in FIG. 19.

In the print processing, the image processing part 14 examines whetherthe document is stored in the document storage part 12 in accordancewith the print job stored in the print queue (step S31). When thedocument exists (step S31/YES), one document is taken out of the printqueue (step S32). In general, the document having arrived at the printqueue earliest is taken out. Here, according to the information ofdocuments, when the sheet with a certain size is used up, the documentwith the size is skipped, and a document which can be outputted at thattime point may be taken out.

The document taken out is subjected to output image formation by colorseparation for output and page layout processing (step S33). Theinformation of the number of pages obtained by the formation of theoutput image is sent to the distribution goodness-of-fittest device 1(step S34), and then, print out or the like is performed by the outputpart 15 so that the output image is outputted (step S35). After thedocument image is outputted, the procedure returns again to the step ofexamining whether a print job exists in the queue, and when the printjob runs out, the processing is ended.

Next, a calculation procedure of steady data performed by the arithmeticpart 17 will be described with reference to a flowchart shown in FIG.20.

The calculation of the steady data is performed at the supply timing ofa sheet in the arithmetic part 17. First, a time from the former supplytime is obtained, and the past average supply time interval is updated(step S41). Subsequently, the number of documents from the timing whenthe sheets are supplied last time and the number of pages of thedocuments are added to the past cumulative value (steps S42 and 43), andthe average number of cumulative documents is calculated (step S44).Further, the storage time of the documents into the document storagepart 12 is counted (step S45), an average number of documents in eachtime and the average number of documents for each day of the week arecalculated (steps S46, S47 and S48), and they are recorded as expectedfrequency data for a test into the data storage part 2 (step S49).

Next, a procedure of page number counting performed in the arithmeticpart 17 will be described with reference to a flowchart shown in FIG.21. The arithmetic part 17 counts the number of documents after supplyand the number of pages on the basis of the page number data sent fromthe image processing part 14 (steps S51 and S52), calculates an averageof the number of pages after the supply, and records it (steps S53 andS54) Next, a procedure of arrival time recording performed in thearithmetic part 17 will be described with reference to a flowchart shownin FIG. 22. In the arithmetic part 17, on the basis of the storagecompletion time sent from the document storage part 12, the arrivalnumbers at the corresponding times and on the corresponding days arecounted (step S61) and are recorded (step S62).

Next, a procedure of a supply judgment made in the arithmetic part 17will be described with reference to a flowchart shown in FIG. 23. Thearithmetic part 17 performs a supply judgment processing described belowin an idling state in which a processing is not performed and betweenoutput processing of the respective documents. First, it is judgedwhether the time is a separation time such as 20th hour, 30th hour or20th day (step S71), and in the case of the time other than theseparation time (step S71/NO), the processing is ended. The separationtime here is the timing when the test is performed, and since [number ofclassifications (number of counting sections) m×10] or more data arerequired in order to improve the accuracy of the test, it is assumedthat the condition of 10 multiplied by the number of classifications(number of classifications≧2) is satisfied. In this embodiment, both thehour and the day are made objects, and there are a test timing on thehour and a test timing on the day.

Subsequently, the distribution goodness-of-fit test device 1 iscontrolled and the distribution goodness-of-fit test processing isperformed (step S72). In the case where there is a significantdifference by the distribution goodness-of-fit test processing (stepS73/YES), since the arrival rate is different from that of the steadystate, as an average consumption rate, an average arrival rate×theaverage number of document pages from the last supply is obtained, aratio thereof to an average arrival rate×the average number of documentpages in the steady state is calculated (step S78) an average supplytime interval is multiplied by this, its result is added to the lastsupply time, and the final result is made an expected supply time (stepS75).

In the case where there is no significant difference as a result of thetest (step S73/NO), the arrival rate is assumed to be the same as thatof the steady state, the ratio of the present average number of documentpages to the average number of document pages of the steady state iscalculated (step S74), an average supply time interval is multiplied bythis, its result is added to the last supply time, and the final resultis made an expected supply time (step S75). This expected supply time iscompared with the present time, and when a remaining time is shorterthan, for example, 1 week (step S76/YES), a warning of an expected timewhen supply is required and the like is issued through a panel of thenotification part 16, and the user is urged to prepare (step S77).

SPECIFIC EXAMPLE 10

A specific example of the foregoing distribution goodness-of-fit test ofstep S72 will be described. TABLE 23 number cumulative of pastprobability cumulative outputs average (k − 1) probability (k)classification 1 classification 2 classification 3 1 1 7.5 0.8691606540.92545308 0 0 1 2 4 4.6 0.333956325 0.522669677 0 1 0 3 0 2.7 00.06213403 1 0 0 4 3 2.3 0.602699718 0.80442588 0 0.3170979320.682902068 5 2 2.8 0.236327697 0.47676273 0.403458827 0.596541173 0 6 23.2 0.175807333 0.387233126 0.745065198 0.254934802 0 7 3 3.6 0.309757470.523582864 0.110257547 0.889742453 0 8 5 5.5 0.366938354 0.539224429 01 0 9 6 7.3 0.273448721 0.41741302 0.415968491 0.584031509 0 10 1 3.30.038234436 0.163032352 1 0 0 11 6 7.5 0.250502744 0.3894254990.596234859 0.403765141 0 12 7 4.6 0.82461197 0.909251438 0 0 0 13 2 2.70.254052001 0.500863353 0.32122239 0.67877761 0 14 1 2.3 0.1028049150.33678089 0.98569842 0.01430158 0 15 4 2.8 0.698721965 0.852399349 0 01 16 2 3.2 0.175807333 0.387233126 0.745065198 0.254934802 0 17 1 3.60.02841757 0.129605366 1 0 0 18 7 5.5 0.695431365 0.816827164 0 0 1 1913 7.3 0.966723209 0.98375412 0 0 1 20 3 3.3 0.366703716 0.588299429 0 10 21 9 7.5 0.673164923 0.785688293 0 0 1 22 3 4.6 0.1680487970.333956325 0.996244948 0.003755052 0 23 3 2.7 0.500863353 0.720571589 00.754652246 0.245347754 24 3 2.3 0.602699718 0.80442588 0 0.3170979320.682902068 25 1 2.8 0.062695189 0.2363227697 1 0 0 26 3 3.2 0.3872331260.610295021 0 1 0 27 4 3.6 0.523582864 0.713927433 0 0.7517094050.248290595 28 6 5.6 0.539224429 0.695431365 0 0.815855179 0.18414482129 10 7.3 0.807461372 0.8850986685 0 0 1 30 2 3.3 0.1630323520.366703716 0.836155747 0.163844253 0

TABLE 24 expected classification observed frequency frequency (Xi −n/m)² ÷ (n/m) 1 10.15537162 10 0.002414 2 10.80104107 10 0.064167 39.043587307 10 0.091473 total 30 30 0.158053238

Table 23 shows an example of data in which the number of outputs in aworking time of a printer is compared with a past average value as asteady state and a test is performed. As shown in Table 24, as a resultof the test, because of χ²(3−1, 0.05)=5.991476357>0.158053238, it isunderstood that the number of outputs well matches the inhomogeneousPoisson process. Thus, in this embodiment, the supply timing iscalculated by only the ratio of the average number of pages perdocument.

Incidentally, in the case of a printer having plural sheet trays, astatistic is obtained for each sheet and a test is performed, so that itis possible to distinguish supply timings for respective sheets.Similarly, in addition to the sheets, also with respect to toner or ink,a statistic is obtain for each color, a test is performed, and a warningof each supply timing can be issued. Besides, when these are notified toa supplier through a network, it is also easy to perform a service ofdelivery at a timing in conformity with a use state.

Embodiment 4

Next, embodiment 4 will be described. This embodiment is an embodimentof a distribution goodness-of-fittest program, and the processing ofdistribution goodness-of-fit test is performed by the processing of acomputer device 30 shown in FIG. 24. As shown in FIG. 24, the computerdevice 30 includes a CPU 31, a ROM 32, a RAM 33, an operation part 34,an I/F part 35 and the like. The distribution goodness-of-fit testprogram is recorded in the ROM 32, and the CPU 31 reads out this programfrom the ROM 32 and executes the processing. The CPU 31 reads outmeasurement data and statistical data recorded in the RAM 33 andperforms the distribution goodness-of-fit test processing in accordancewith the flowchart shown in FIG. 5. In this way, also in thisembodiment, the same effects as the foregoing respective embodiments canbe obtained.

Incidentally, the foregoing embodiments are preferred embodiments of thepresent invention. However, the invention is not limited to these, butcan be variously modified within the scope not departing from the gistof the invention. For example, in the foregoing embodiments, althoughthe discrete distribution such as the inhomogeneous Poisson distributionand the binomial distribution has been described, even in a continuousdistribution, when division into sections is performed and totalizationis made, it can be treated as the discrete distribution, and therefore,the invention can be sufficiently applied.

As described above, some embodiments of the present invention areoutlined below:

In the distribution goodness-of-fit test device of the presentinvention, the counting section determination unit determines thecounting sections so that the widths of the respective counting sectionshave equal probabilities on the estimated probability distribution.

According to this invention, the counting sections are determined sothat the widths of the respective counting sections have the equalprobabilities on the estimated probability distribution. Accordingly,even if a distribution is such that a probability distribution variestemporally, the distribution is converted into a uniform distribution,and a goodness-of-fit test can be performed. Thus, the test possiblerange of the goodness-of-fit test can be widened.

In the distribution goodness-of-fit test device of the presentinvention, the counting section determination unit determines aspecified division number m corresponding to the number of the measureddata, divides the estimated probability distribution into equalprobability sections each having a probability of 1/m, and determinesthe widths of the respective counting sections.

According to this invention, the division number m is determinedaccording to the number of the measured data. In the case where thenumber of the data obtained by measurement is small, the division numberis changed so that the test with high accuracy can be performed.Besides, the estimated probability distribution is divided into theequal probability sections each having the probability of 1/m and thewidths of the respective counting sections are determined. Accordingly,even if the distribution is such that the probability distribution ischanged temporally, the distribution is converted into the uniformdistribution and the goodness-of-fit test can be performed. Thus, thetest possible range of the goodness-of-fit test can be widened.

In the distribution goodness-of-fit test device of the presentinvention, in the case where a discrete value of the measurement data isk, the data counting unit judges the equal probability sections in whicha cumulative probability of from 0 to k is included, and classifies themeasurement data into the counting sections.

According to this invention, the cumulative probability of the discretevalue is obtained, so that the measurement data can be classified intothe counting sections.

In the distribution goodness-of-fit test device of the presentinvention, it is assumed that the measured data follows pluralprobability distributions, and the counting section determination unitchanges the widths of the counting sections for each of the assumedprobability distributions.

According to this invention, the widths of the counting sections arechanged for each of the assumed probability distributions and thecounting of data is performed, so that it is possible to cause theprobabilities of the respective counting sections on the estimatedprobability distribution to become equal probabilities. Accordingly,even if the distribution is such that the probability distributionvaries temporally, the distribution is converted into the uniformdistribution and the goodness-of-fit test can be performed. Thus, thetest possible range of the goodness-of-fit test can be widened.

In the distribution goodness-of-fit test device of the presentinvention, in a case where a discrete value of the measurement dataextends over plural counting sections, the data counting unit dividesthe data into plural counting sections and counts.

According to this invention, in the case where the probability of takingthe discrete value of the measurement data extends over the pluralcounting sections, the data is divided into the plural counting sectionsand is counted, so that an error can be made small.

In the distribution goodness-of-fit test device of the presentinvention, in a case where a discrete value of the measurement dataextends over plural counting sections, the data counting unit obtains arate at which a probability of taking the discrete value of themeasurement data is included in the equal probability section, dividesthe measurement data into the plural counting sections, and counts.

According to this invention, the measurement data is divided into theplural counting sections according to the rate at which the probabilityof taking the discrete value of the measurement data is included in therespective counting sections and is counted. Accordingly, themeasurement data can be divided into the plural counting sections withhigh accuracy, and the error can be made further small.

In the distribution goodness-of-fit test device of the presentinvention, the goodness-of-fit test unit tests the counted data by achi-square goodness-of-fit test.

The counted data is tested by the chi-square goodness-of-fit test, sothat the test can be performed using a generally used test method.

In the distribution goodness-of-fit test device of the presentinvention, the estimated probability distribution is a probabilitydistribution varying temporally.

Even if the estimated probability distribution is the probabilitydistribution varying temporally, the widths of the counting sections aredetermined so that the widths of the respective counting sections haveequal probabilities on the estimated probability distribution, andtherefore, the distribution is converted into a uniform distribution andthe goodness-of-fit test can be performed.

In the distribution goodness-of-fit test device of the presentinvention, the estimated probability distribution follows ainhomogeneous Poisson process.

Even if the estimated probability distribution follows the inhomogeneousPoisson process, the goodness-of-fit test can be performed.

In the distribution goodness-of-fit test method of the presentinvention, the counting section determination step determines thecounting sections so that the widths of the respective counting sectionshave equal probabilities on the estimated probability distribution.

According to this invention, the counting sections are determined sothat the widths of the respective counting sections have the equalprobabilities on the estimated probability distribution. Accordingly,even if a distribution is such that a probability distribution variestemporally, the distribution is converted into a uniform distribution,and a goodness-of-fit test can be performed. Thus, the test possiblerange of the goodness-of-fit test can be widened.

In the distribution goodness-of-fit test method of the presentinvention, it is assumed that the measured data follows pluralprobability distributions, and the counting section determination stepdetermines the widths of the counting sections for each of the assumedprobability distributions.

According to this invention, the widths of the counting sections arechanged for each of the assumed probability distributions and thecounting of the data is performed, so that it is possible to cause theprobabilities of the respective counting sections on the estimatedprobability distribution to become equal probabilities. Accordingly,even if the distribution is such that the probability distributionvaries temporally, the distribution is converted into the uniformdistribution and the goodness-of-fit test can be performed. Thus, thetest possible range of the goodness-of-fit test can be widened.

In the distribution goodness-of-fit test method of the presentinvention, in a case where a discrete value of the measurement dataextends over plural counting sections, the data counting step dividesthe data into plural counting sections and counts.

According to this invention, in the case where the probability of takingthe discrete value of the measurement data extends over the pluralcounting sections, the data is divided into the plural counting sectionsand is counted, so that an error can be made small.

In the storage medium readable by the computer and storing thedistribution goodness-of-fit test program of the present invention, thecounting section determination step determines the counting sections sothat the widths of the respective counting sections have equalprobabilities on the estimated probability distribution.

According to this invention, the counting sections are determined sothat the widths of the respective counting sections have the equalprobabilities on the estimated probability distribution. Accordingly,even if a distribution is such that a probability distribution variestemporally, the distribution is converted into a uniform distribution,and a goodness-of-fit test can be performed. Thus, the test possiblerange of the goodness-of-fit test can be widened.

In the storage medium readable by the computer and storing thedistribution goodness-of-fit test program of the present invention, itis assumed that the measured data follows plural probabilitydistributions, and the counting section determination step determinesthe widths of the counting sections for each of the assumed probabilitydistributions.

According to this invention, the widths of the counting sections arechanged for each of the assumed probability distributions and thecounting of the data is performed, so that it is possible to cause theprobabilities of the respective counting sections on the estimatedprobability distribution to become equal probabilities. Accordingly,even if the distribution is such that the probability distributionvaries temporally, the distribution is converted into the uniformdistribution and the goodness-of-fit test can be performed. Thus, thetest possible range of the goodness-of-fit test can be widened.

In the storage medium readable by the computer and storing thedistribution goodness-of-fit test program of the present invention, in acase where a discrete value of the measurement data extends over pluralcounting sections, the data counting step divides the data into pluralcounting sections and counts.

According to this invention, in the case where the probability of takingthe discrete value of the measurement data extends over the pluralcounting sections, the data is divided into the plural counting sectionsand is counted, so that an error can be made small.

The foregoing description of the embodiments of the present inventionhas been provided for the purposes of illustration and description. Itis not intended to be exhaustive or to limit the invention to theprecise forms disclosed. Obviously, many modifications and variationswill be apparent to practitioners skilled in the art. The embodimentswere chosen and described in order to best explain the principles of theinvention and its practical applications, thereby enabling othersskilled in the art to understand the invention for various embodimentsand with the various modifications as are suited to the particular usecontemplated. It is intended that the scope of the invention be definedby the following claims and their equivalents.

The entire disclosure of Japanese Patent Application No. 2004-183148filed on Jun. 21, 2004 including specification, claims, drawings andabstract is incorporated herein by reference in its entirety.

1. A distribution goodness-of-fit test device for testing whethermeasured data matches an estimated probability distribution, comprising:a counting section determination unit that determines according to thenumber of the measured data, widths of counting sections for countingthe measured data; a counting unit that counts the numbers of data inthe respective determined counting sections; and a goodness-of-fit testunit that performs a goodness-of-fit test based on the numbers of datain the respective counting sections.
 2. A distribution goodness-of-fittest device for testing whether measured data matches an estimatedprobability distribution, comprising: a counting section determinationunit that determines, according to the estimated probabilitydistribution, widths of counting sections for counting the measureddata; a counting unit that counts the numbers of data in the respectivedetermined counting sections; and a goodness-of-fit test unit thatperforms a goodness-of-fit test based on the numbers of data in therespective counting sections.
 3. A distribution goodness-of-fit testdevice for testing whether measured data matches an estimatedprobability distribution, comprising: a counting section determinationunit that determines, according to the estimated probabilitydistribution and the number of the measured data, widths of countingsections for counting the measured data; a counting unit that counts thenumbers of data in the respective determined counting sections; and agoodness-of-fit test unit that performs a goodness-of-fit test based onthe numbers of data in the respective counting sections.
 4. Adistribution goodness-of-fit test device according to claim 1, whereinthe counting section determination unit determines the countingsections, and the widths of the respective counting sections have equalprobabilities on the estimated probability distribution.
 5. Adistribution goodness-of-fit test device according to claim 1, whereinthe counting section determination unit determines a specified divisionnumber m corresponding to the number of the measured data, divides theestimated probability distribution into equal probability sections eachhaving a probability of 1/m, and determines the widths of the respectivecounting sections.
 6. A distribution goodness-of-fit test deviceaccording to claim 5, wherein when a discrete value of the measurementdata is k, the counting unit judges the equal probability sections inwhich a cumulative probability of from 0 to k is included, andclassifies the measurement data into the counting sections.
 7. Adistribution goodness-of-fit test device according to claim 1, whereinit is assumed that the measured data follows a plurality of probabilitydistributions, and the counting section determination unit determinesthe widths of the counting sections for each of the assumed probabilitydistributions.
 8. A distribution goodness-of-fit test device accordingto claim 1, wherein when a discrete value of the measurement dataextends over a plurality of counting sections, the counting unit dividesthe data into a plurality of counting sections and counts.
 9. Adistribution goodness-of-fit test device according to claim 6, whereinwhen a discrete value of the measurement data extends over a pluralityof counting sections, the counting unit obtains a rate at which aprobability of taking the discrete value of the measurement data isincluded in the equal probability sections, divides the measurement datainto the a plurality of counting sections, and counts.
 10. Adistribution goodness-of-fit test device according to claim 1, whereinthe goodness-of-fit test unit tests the counted data by a chi-squaregoodness-of-fit test.
 11. A distribution goodness-of-fit test deviceaccording to claim 1, wherein the estimated probability distribution isa probability distribution varying temporally.
 12. A distributiongoodness-of-fit test device according to claim 1, wherein the estimatedprobability distribution follows a inhomogeneous Poisson process.
 13. Aconsumable goods supply timing judgment device comprising: adistribution goodness-of-fit test device for testing whether measureddata matches an estimated probability distribution, further comprising:a counting section determination unit that determines according to thenumber of the measured data, widths of counting sections for countingthe measured data; a counting unit that counts the numbers of data inthe respective determined counting sections; and a goodness-of-fit testunit that performs a goodness-of-fit test based on the numbers of datain the respective counting sections; wherein the goodness-of-fit testfrom a measured consumption rate of consumable goods per unit time andan average value of a past consumption rate of the consumable goods perunit time, and the consumable goods supply timing judgment devicefurther comprising: a control unit that calculates an estimated supplytime from a ratio of a present consumption rate of the consumable goodsto a past consumption rate of the consumable goods and notifying theestimated supply time when a difference between the present consumptionrate and the past consumption rate is judged by the goodness-of-fittest.
 14. An image forming device comprising: a consumable goods supplytiming judgment device comprising: a distribution goodness-of-fit testdevice for testing whether measured data matches an estimatedprobability distribution, further comprising: a counting sectiondetermination unit that determines according to the number of themeasured data, widths of counting sections for counting the measureddata; a counting unit that counts the numbers of data in the respectivedetermined counting sections; and a goodness-of-fit test unit thatperforms a goodness-of-fit test based on the numbers of data in therespective counting sections; wherein the goodness-of-fit test from ameasured consumption rate of consumable goods per unit time and anaverage value of a past consumption rate of the consumable goods perunit time, and the consumable goods supply timing judgment devicefurther comprising: a control unit that calculates an estimated supplytime from a ratio of a present consumption rate of the consumable goodsto a past consumption rate of the consumable goods and notifying theestimated supply time when a difference between the present consumptionrate and the past consumption rate is judged by the goodness-of-fittest.
 15. A distribution goodness-of-fit test method for testing whethermeasured data matches an estimated probability distribution, comprising:determining, according to the estimated probability distribution and thenumber of the measured data, widths of counting sections for countingthe measured data; counting the numbers of data in the respectivedetermined counting sections; and performing a goodness-of-fit testbased on the numbers of data in the respective counting sections.
 16. Adistribution goodness-of-fit test method according to claim 15, whereinthe widths of the respective counting sections have equal probabilitieson the estimated probability distribution by determining the countingsections.
 17. A distribution goodness-of-fit test method according toclaim 15, wherein it is assumed that the measured data follows aplurality of probability distributions, and the widths of the countingsections are determined based on each of the assumed probabilitydistributions.
 18. A distribution goodness-of-fit test method accordingto claim 15, wherein when a discrete value of the measurement dataextends over a plurality of counting sections, the data is divided by aplurality of counting sections and being counted.
 19. A storage mediumreadable by a computer and storing a distribution goodness-of-fit testprogram of instructions executable by the computer to perform a functionfor testing whether measured data matches an estimated probabilitydistribution, the function comprising: determining, according to theestimated probability distribution and the number of the measured data,widths of counting sections for counting the measured data; counting thenumbers of data in the respective determined counting sections; andperforming a goodness-of-fit test using the numbers of data in therespective counting sections.
 20. A storage medium readable by acomputer and storing a distribution goodness-of-fit test programaccording to claim 19, wherein the widths of the respective countingsections have equal probabilities on the estimated probabilitydistribution by determining the counting sections.
 21. A storage mediumreadable by a computer and storing a distribution goodness-of-fit testprogram according to claim 19, wherein it is assumed that the measureddata follows a plurality of probability distributions, and the widths ofthe counting sections are determined based on each of the assumedprobability distributions.
 22. A storage medium readable by a computerand storing a distribution goodness-of-fit test program according toclaim 19, wherein when a discrete value of the measurement data extendsover a plurality of counting sections, the data is divided by aplurality of counting sections and being counted.